Subgroups — #34204729

Order 20224.

462 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₇₉⋊C₁₆	C₂	1	2	yes	maximal, complemented
2	C₂×C₂²⋊(C₇₉⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
3	C₈×C₂²⋊Dic₇₉	C₂	1	1	yes	maximal, characteristic
4	#23761477	C₂	1	2	yes	maximal, complemented
5	C₂³×C₁₆×C₇₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₇₉⋊C₁₆	C₂²	1	1	yes	characteristic
7	C₂²×C₇₉⋊C₁₆	C₂²	1	1	yes	characteristic
8	C₂×C₂²⋊(C₇₉⋊C₈)	C₄	1	1	yes	characteristic
9	C₂×C₈×Dic₇₉	C₂²	1	2	yes	—
10	C₄×C₂²⋊Dic₇₉	C₄	1	1	yes	characteristic
11	Dic₇₉⋊C₁₆	—	2	4	no	—
12	C₂²⋊(C₇₉⋊C₁₆)	—	2	2	no	—
13	#19098317	C₄	1	2	yes	—
14	C₂³×C₈×C₇₉	C₂²	1	1	yes	characteristic
15	C₂²×C₁₆×C₇₉	—	2	2	no	—
16	C₂²×C₁₆×C₇₉	—	2	4	no	—
17	C₂²×C₁₆×C₇₉	C₂²	1	2	yes	—
18	C₂²×C₇₉⋊C₈	C₂×C₄	1	2	yes	—
19	C₂×C₂²⋊Dic₇₉	C₈	1	1	yes	characteristic
20	C₂×C₂²⋊Dic₇₉	C₈	1	1	yes	characteristic
21	C₂×C₄×Dic₇₉	C₂×C₄	1	2	yes	—
22	C₂×C₇₉⋊C₁₆	—	2	2	no	—
23	C₂×C₇₉⋊C₁₆	—	2	1	no	—
24	C₂×C₇₉⋊C₁₆	—	2	1	no	—
25	C₂×C₇₉⋊C₁₆	—	2	2	no	—
26	C₂²⋊(C₇₉⋊C₈)	—	2	2	no	—
27	C₈×Dic₇₉	—	2	2	no	—
28	C₈×Dic₇₉	—	2	2	no	—
29	C₇₉⋊#95	C₈	1	2	yes	—
30	C₂³×C₄×C₇₉	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₈×C₇₉	—	2	2	no	—
32	C₂²×C₈×C₇₉	—	2	2	no	—
33	C₂²×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₈×C₇₉	D₄	1	2	yes	—
35	C₂²×C₈×C₇₉	Q₈	1	2	yes	—
36	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
37	C₂×C₁₆×C₇₉	—	2	8	no	—
38	C₂×C₁₆×C₇₉	—	2	2	no	—
39	C₂×C₁₆×C₇₉	—	2	2	no	—
40	C₂×C₁₆×C₇₉	D₄	1	4	yes	—
41	C₂²×Dic₇₉	C₂×C₈	1	2	yes	—
42	C₂²×Dic₇₉	C₂×C₈	1	2	yes	—
43	C₂×C₇₉⋊C₈	—	2	2	no	—
44	C₂×C₇₉⋊C₈	—	2	2	no	—
45	C₂×C₇₉⋊C₈	—	2	2	no	—
46	C₂²⋊Dic₇₉	—	2	2	no	—
47	C₂²⋊Dic₇₉	C₁₆	1	4	yes	complemented
48	C₄×Dic₇₉	—	2	2	no	—
49	C₄×Dic₇₉	—	2	2	no	—
50	C₇₉⋊C₁₆	—	4	1	no	—
51	C₇₉⋊C₁₆	—	4	1	no	—
52	C₂⁴×C₇₉	C₂×C₈	1	1	yes	characteristic
53	C₂²×C₄×C₇₉	—	2	2	no	—
54	C₂²×C₄×C₇₉	—	2	2	no	—
55	C₂²×C₄×C₇₉	C₄⋊C₄	1	2	yes	—
56	C₂²×C₄×C₇₉	C₂×C₈	1	1	yes	characteristic
57	C₂²×C₄×C₇₉	M₁₆	1	2	yes	—
58	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₈×C₇₉	—	2	2	no	—
60	C₂×C₈×C₇₉	—	2	4	no	—
61	C₂×C₈×C₇₉	—	2	2	no	—
62	C₂×C₈×C₇₉	—	2	2	no	—
63	C₂×C₈×C₇₉	C₄⋊C₄	1	2	yes	—
64	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
65	C₂×C₈×C₇₉	C₂×D₄	1	2	yes	—
66	C₂×C₈×C₇₉	C₂×Q₈	1	1	yes	characteristic
67	C₂×C₈×C₇₉	#41	1	2	yes	—
68	C₁₆×C₇₉	—	2	4	no	—
69	C₂³×C₇₉	—	2	2	no	—
70	C₂³×C₇₉	—	2	2	no	—
71	C₂³×C₇₉	C₄⋊C₈	1	2	yes	—
72	C₂³×C₇₉	C₂×C₁₆	1	2	yes	—
73	C₂³×C₇₉	M₃₂	1	2	yes	—
74	C₂³×C₇₉	C₂²×C₈	1	1	yes	characteristic
75	C₂×C₄×C₇₉	—	2	2	no	—
76	C₂×C₄×C₇₉	—	2	2	no	—
77	C₂×C₄×C₇₉	—	2	2	no	—
78	C₂×C₄×C₇₉	—	2	4	no	—
79	C₂×C₄×C₇₉	C₄⋊C₈	1	2	yes	—
80	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
81	C₂×C₄×C₇₉	C₄×D₄	1	2	yes	—
82	C₂×C₄×C₇₉	C₂×M₁₆	1	1	yes	characteristic
83	C₂×C₄×C₇₉	#131	1	2	yes	—
84	C₂×Dic₇₉	—	2	2	no	—
85	C₂×Dic₇₉	—	2	2	no	—
86	C₂×Dic₇₉	—	2	2	no	—
87	C₂×Dic₇₉	—	2	2	no	—
88	C₂×Dic₇₉	—	2	2	no	—
89	C₂×Dic₇₉	C₂×C₁₆	1	4	yes	complemented
90	C₈×C₇₉	—	2	2	no	—
91	C₈×C₇₉	#120	1	1	yes	characteristic
92	C₈×C₇₉	C₂²⋊Q₈	1	2	yes	—
93	C₈×C₇₉	#123	1	1	yes	characteristic
94	C₇₉⋊C₈	—	4	2	no	—
95	C₂²×C₇₉	—	2	2	no	—
96	C₂²×C₇₉	—	2	2	no	—
97	C₂²×C₇₉	—	2	4	no	—
98	C₂²×C₇₉	—	2	4	no	—
99	C₂²×C₇₉	C₄⋊C₁₆	1	4	yes	complemented
100	C₂²×C₇₉	C₂×C₄⋊C₈	1	1	yes	characteristic
101	C₂²×C₇₉	C₈×D₄	1	2	yes	—
102	C₂²×C₇₉	C₂²×C₁₆	1	1	yes	characteristic
103	C₂²×C₇₉	C₂×M₃₂	1	1	yes	characteristic
104	C₂²×C₇₉	#504	1	2	yes	—
105	C₄×C₇₉	—	2	2	no	—
106	C₄×C₇₉	#385	1	1	yes	characteristic
107	C₄×C₇₉	#424	1	1	yes	characteristic
108	C₄×C₇₉	#435	1	2	yes	—
109	Dic₇₉	—	2	4	no	—
110	Dic₇₉	—	4	2	no	—
111	#12912	—	79	1	no	maximal, Sylow
112	C₂×C₇₉	—	2	4	no	—
113	C₂×C₇₉	#1850	1	1	yes	characteristic
114	C₂×C₇₉	C₂×C₄⋊C₁₆	1	1	yes	characteristic
115	C₂×C₇₉	#2151	1	1	yes	characteristic
116	C₂×C₇₉	D₄×C₁₆	1	2	yes	—
117	C₂×C₇₉	#2168	1	2	yes	—
118	C₂³×C₁₆	D₇₉	1	1	yes	characteristic
119	C₂×C₄⋊C₁₆	—	79	2	no	—
120	C₂×C₂²⋊C₁₆	—	79	1	no	—
121	C₈×C₂²⋊C₄	—	79	1	no	—
122	#1374	—	79	2	no	—
123	C₇₉	#12912	1	1	yes	characteristic, complemented, Sylow
124	C₂³×C₈	C₂×D₇₉	1	1	yes	characteristic
125	C₂²×C₁₆	—	2	2	no	—
126	C₂²×C₁₆	—	2	4	no	—
127	C₂²×C₁₆	—	79	1	no	—
128	C₂²×C₁₆	—	79	1	no	—
129	C₂²×C₁₆	C₂×D₇₉	1	2	yes	—
130	C₂×C₂²⋊C₈	—	79	1	no	—
131	C₂×C₄×C₈	—	79	2	no	—
132	C₄×C₂²⋊C₄	—	79	1	no	—
133	C₄⋊C₁₆	—	158	4	no	—
134	C₂²⋊C₁₆	—	158	2	no	—
135	#336	—	79	2	no	—
136	C₂³×C₄	C₄×D₇₉	1	1	yes	characteristic
137	C₂²×C₈	—	2	2	no	—
138	C₂²×C₈	—	2	2	no	—
139	C₂²×C₈	—	79	2	no	—
140	C₂²×C₈	Dic₁₅₈	1	2	yes	—
141	C₂²×C₈	C₄×D₇₉	1	1	yes	characteristic
142	C₂²×C₈	C₂²⋊D₇₉	1	2	yes	—
143	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
144	C₂×C₂²⋊C₄	—	79	1	no	—
145	C₂×C₂²⋊C₄	—	79	1	no	—
146	C₂×C₄²	—	79	2	no	—
147	C₂×C₁₆	—	2	8	no	—
148	C₂×C₁₆	—	2	2	no	—
149	C₂×C₁₆	—	2	2	no	—
150	C₂×C₁₆	—	158	1	no	—
151	C₂×C₁₆	—	158	1	no	—
152	C₂×C₁₆	—	158	2	no	—
153	C₂×C₁₆	—	158	2	no	—
154	C₂×C₁₆	C₂²⋊D₇₉	1	4	yes	—
155	C₂²⋊C₈	—	158	2	no	—
156	C₄×C₈	—	158	2	no	—
157	C₄×C₈	—	158	2	no	—
158	#95	—	79	2	no	—
159	C₂⁴	C₈×D₇₉	1	1	yes	characteristic
160	C₂²×C₄	—	2	2	no	—
161	C₂²×C₄	—	2	2	no	—
162	C₂²×C₄	—	79	2	no	—
163	C₂²×C₄	—	79	2	no	—
164	C₂²×C₄	Dic₇₉⋊C₄	1	2	yes	—
165	C₂²×C₄	C₈×D₇₉	1	1	yes	characteristic
166	C₂²×C₄	#11929607	1	2	yes	—
167	C₂²×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
168	C₂×C₈	—	2	2	no	—
169	C₂×C₈	—	2	2	no	—
170	C₂×C₈	—	2	4	no	—
171	C₂×C₈	—	2	2	no	—
172	C₂×C₈	—	158	2	no	—
173	C₂×C₈	—	158	2	no	—
174	C₂×C₈	—	158	2	no	—
175	C₂×C₈	Dic₇₉⋊C₄	1	2	yes	—
176	C₂×C₈	C₂×C₂²⋊D₇₉	1	2	yes	—
177	C₂×C₈	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
178	C₂×C₈	C₂×Dic₁₅₈	1	1	yes	characteristic
179	C₂×C₈	#11929624	1	2	yes	—
180	C₂²⋊C₄	—	79	4	no	—
181	C₂²⋊C₄	—	158	2	no	—
182	C₄²	—	158	2	no	—
183	C₄²	—	158	2	no	—
184	C₁₆	—	2	4	no	—
185	C₁₆	—	316	1	no	—
186	C₁₆	—	316	1	no	—
187	C₂³	—	2	2	no	—
188	C₂³	—	2	2	no	—
189	C₂³	Dic₇₉⋊C₈	1	2	yes	—
190	C₂³	C₁₆×D₇₉	1	2	yes	—
191	C₂³	#14579100	1	2	yes	—
192	C₂³	C₂×C₈×D₇₉	1	1	yes	characteristic
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	2	4	no	—
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	79	4	no	—
198	C₂×C₄	—	158	2	no	—
199	C₂×C₄	—	158	2	no	—
200	C₂×C₄	—	158	2	no	—
201	C₂×C₄	—	158	2	no	—
202	C₂×C₄	—	158	2	no	—
203	C₂×C₄	Dic₇₉⋊C₈	1	2	yes	—
204	C₂×C₄	C₂×Dic₇₉⋊C₄	1	1	yes	characteristic
205	C₂×C₄	C₄×C₂²⋊D₇₉	1	2	yes	—
206	C₂×C₄	C₂×#11929607	1	1	yes	characteristic
207	C₂×C₄	#14579167	1	2	yes	—
208	C₈	—	2	2	no	—
209	C₈	—	316	2	no	—
210	C₈	#14579130	1	1	yes	characteristic
211	C₈	#14579140	1	2	yes	—
212	C₈	#14579144	1	1	yes	characteristic
213	C₂²	—	2	2	no	—
214	C₂²	—	2	2	no	—
215	C₂²	—	2	4	no	—
216	C₂²	—	2	4	no	—
217	C₂²	Dic₇₉⋊C₁₆	1	4	yes	complemented
218	C₂²	C₂×Dic₇₉⋊C₈	1	1	yes	characteristic
219	C₂²	C₈×C₂²⋊D₇₉	1	2	yes	—
220	C₂²	C₂×C₁₆×D₇₉	1	1	yes	characteristic
221	C₂²	C₂×#14579100	1	1	yes	characteristic
222	C₂²	#19098979	1	2	yes	—
223	C₄	—	2	2	no	—
224	C₄	—	316	2	no	—
225	C₄	—	158	4	no	—
226	C₄	#19098436	1	1	yes	characteristic
227	C₄	#19098589	1	1	yes	characteristic
228	C₄	#19098634	1	2	yes	—
229	C₂	—	2	4	no	—
230	C₂	#23763578	1	1	yes	characteristic
231	C₂	C₂×Dic₇₉⋊C₁₆	1	1	yes	characteristic
232	C₂	#23765158	1	1	yes	characteristic
233	C₂	C₁₆×C₂²⋊D₇₉	1	2	yes	—
234	C₂	#23765225	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₇₉
commutator	C₂²×C₇₉	C₂²×C₁₆
Fitting	C₂³×C₁₆×C₇₉	C₂
socle	C₂³×C₇₉	C₂²×C₈
prime radical	C₂⁴×C₇₉	C₂×C₈
cosocle	C₂²×C₈×C₇₉	C₂³
hypercenter	C₂³×C₁₆	D₇₉
hypocenter	C₇₉	#12912
LCS	G ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂²×C₈ ◁ C₂³×C₁₆

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₂×Dic₇₉⋊C₁₆	C₂	semidirect
4. #23761477	C₂	semidirect
47. C₂²⋊Dic₇₉	C₁₆	semidirect
89. C₂×Dic₇₉	C₂×C₁₆	semidirect
99. C₂²×C₇₉	C₄⋊C₁₆	semidirect
123. C₇₉	#12912	semidirect
217. C₂²	Dic₇₉⋊C₁₆	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×Dic₇₉⋊C₁₆	2
C₂×C₂²⋊(C₇₉⋊C₁₆)	2
C₈×C₂²⋊Dic₇₉	2
#23761477	2
C₂³×C₁₆×C₇₉	2
#12912	79